| Exam Board | Edexcel |
|---|---|
| Module | F3 (Further Pure Mathematics 3) |
| Year | 2018 |
| Session | Specimen |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hyperbolic functions |
| Type | Differentiate inverse hyperbolic functions |
| Difficulty | Challenging +1.3 Part (a) requires knowing the derivative of artanh and applying chain rule—straightforward for Further Maths students. Part (b) requires integration by parts with careful algebraic manipulation and evaluation at specific limits involving surds, which is more demanding but follows a standard technique. The 7-mark total and multi-step nature elevate it above average, but it's a recognizable Further Maths exercise without requiring novel insight. |
| Spec | 1.08i Integration by parts4.07f Inverse hyperbolic: logarithmic forms |
| Answer | Marks |
|---|---|
| 5(a) | y (cid:32)artanh(cosx) |
| Answer | Marks | Guidance |
|---|---|---|
| dx 1(cid:16)cos2 x | Correct use of the chain rule | M1 |
| Answer | Marks |
|---|---|
| sin2 x sinx * | A1: Correct completion with no |
| errors | A1 |
| Answer | Marks |
|---|---|
| dx sech2y 1(cid:16)cos2 x | Correct differentiation to obtain a |
| function of x | M1 |
| Answer | Marks |
|---|---|
| sin2 x sinx * | A1: Correct completion with no |
| errors | A1 |
| Answer | Marks |
|---|---|
| dx 2 1(cid:14)cosx (cid:11)1(cid:16)cosx(cid:12)2 | Correct differentiation |
| Answer | Marks |
|---|---|
| x | M1 |
| Answer | Marks |
|---|---|
| * | A1: Correct |
| Answer | Marks |
|---|---|
| errors | A1 |
| Answer | Marks |
|---|---|
| (b) | (cid:179) (cid:179) |
| Answer | Marks |
|---|---|
| M1: Parts in the correct direction A1: Correct expression | M1 A1 |
| Answer | Marks |
|---|---|
| Lower limit need not be shown | M1 |
| Answer | Marks |
|---|---|
| 2 | Use of the logarithmic form of |
| artanh | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| 4 6 2 6 | Cao (oe) | A1 |
| Answer | Marks |
|---|---|
| reverse order. | (5) |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Scheme | Marks |
Question 5:
--- 5(a) ---
5(a) | y (cid:32)artanh(cosx)
dy 1
(cid:32) (cid:117)(cid:16)sinx
dx 1(cid:16)cos2 x | Correct use of the chain rule | M1
(cid:16)sinx (cid:16)1
(cid:32) (cid:32) (cid:32)(cid:16)cosecx
sin2 x sinx * | A1: Correct completion with no
errors | A1
(2)
Alternative 1
dy
tanh y(cid:32)cosx(cid:159)sech2y (cid:32)(cid:16)sinx
dx
dy (cid:16)sinx (cid:16)sinx
(cid:32) (cid:32)
dx sech2y 1(cid:16)cos2 x | Correct differentiation to obtain a
function of x | M1
(cid:16)sinx (cid:16)1
(cid:32) (cid:32) (cid:32)(cid:16)cosecx
sin2 x sinx * | A1: Correct completion with no
errors | A1
(2)
Alternative 2
1 (cid:167)1(cid:14)cosx(cid:183)
artanh(cosx)(cid:32) ln(cid:168) (cid:184)
2 (cid:169)1(cid:16)cosx(cid:185)
(cid:16)sinx(cid:11)1(cid:16)cosx(cid:12)(cid:16)sinx(cid:11)1(cid:14)cosx(cid:12)
dy 1 1(cid:16)cosx
(cid:32) (cid:117) (cid:117)
dx 2 1(cid:14)cosx (cid:11)1(cid:16)cosx(cid:12)2 | Correct differentiation
to obtain a function of
x | M1
(cid:16)2sinx
(cid:32) (cid:32)(cid:16)cosecx
2 (cid:11) 1(cid:16)cos2 x (cid:12)
* | A1: Correct
completion with no
errors | A1
(2)
(b) | (cid:179) (cid:179)
cosxartanh(cid:11)cosx(cid:12)dx(cid:32)sinxartanh(cid:11)cosx(cid:12)(cid:16) sinx(cid:117)(cid:16)cosecxdx
M1: Parts in the correct direction A1: Correct expression | M1 A1
(cid:170) (cid:172) sinxartanh(cid:11)cosx(cid:12)(cid:14)x(cid:186) (cid:188) (cid:83) 6 (cid:32) 1 artanh (cid:167) (cid:168) (cid:168) 3(cid:183) (cid:184) (cid:184) (cid:14) (cid:83) (cid:11) (cid:16)(cid:11)0(cid:12)(cid:12)
0 2 2 6
(cid:169) (cid:185)
M1: Correct use of limits on either part (provided both parts are integrated).
Lower limit need not be shown | M1
1 (cid:167)1(cid:14) 3 (cid:183) (cid:83)
(cid:32) ln(cid:168) 2 (cid:184)(cid:14)
4 1(cid:16) 3 6
(cid:169) (cid:185)
2 | Use of the logarithmic form of
artanh | M1
1 (cid:11) (cid:12) (cid:83) 1 (cid:11) (cid:12) (cid:83)
(cid:32) ln 7(cid:14)4 3 (cid:14) or ln 2(cid:14) 3 (cid:14)
4 6 2 6 | Cao (oe) | A1
The last 2 M marks may be gained in
reverse order. | (5)
(7 marks)
Question | Scheme | MarksGiven that $y = \text{artanh}(\cos x)$
\begin{enumerate}[label=(\alph*)]
\item show that
$$\frac{dy}{dx} = -\text{cosec } x$$
[2]
\item Hence find the exact value of
$$\int_{0}^{\frac{\pi}{4}} \cos x \, \text{artanh}(\cos x) \, dx$$
giving your answer in the form $a \ln\left(b + c\sqrt{3}\right) + d\pi$, where $a$, $b$, $c$ and $d$ are rational numbers to be found.
[5]
\end{enumerate}
\hfill \mbox{\textit{Edexcel F3 2018 Q5 [7]}}